How to Find the Derivative of y=x^y When y is Not Expressed in terms of x?

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Discussion Overview

The discussion revolves around finding the derivative of the equation y = x^y, particularly when y is not expressed explicitly in terms of x. Participants explore various approaches to differentiate the equation and express their thoughts on the implications of the relationship between x and y.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a method involving logarithmic differentiation to find the derivative, leading to the expression y' = (y^2) / (x - xy ln x).
  • Another participant expresses that the provided solution did not meet their expectations, indicating they were looking for a different form of the derivative.
  • A later reply suggests that the original question may have been misunderstood, as the typical expectation is to find the derivative of y with respect to x.
  • One participant clarifies that since y does not equal a function of x, it cannot be expressed in terms of x, proposing that x should instead be expressed in terms of y.
  • Another participant notes that if the equation were x = y^x, it could be expressed as y = x^(1/x), which might simplify the differentiation process.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the approach to take for differentiation, with multiple competing views on how to interpret the relationship between x and y and the expectations for the derivative.

Contextual Notes

There are limitations regarding the assumptions made about the relationship between x and y, as well as the implications of expressing one variable in terms of the other. The discussion reflects varying interpretations of the problem and the methods used to approach it.

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How do you find the derivative of y=x^y ?
 
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[tex]\begin{align*}<br /> y&=x^y \\<br /> \ln y&=y\ln x \\<br /> \frac{y^\prime}{y}&=y^\prime\ln x+\frac{y}{x} \\<br /> y^\prime\left(\frac{1}{y}-\ln x\right)&=\frac{y}{x} \\<br /> y^\prime&=\frac{y^2}{x-xy\ln x}<br /> \end{align*}[/tex]
 
Not quite the answer I was looking for, but thanks any how. I already know how the problem is solved.
 
You asked me how to find the derivative. I showed you a way to do it. What were you looking for?
 
I was looking for something more like this: x'=y^(1/y)*(1/y^2-ln(y)/y^2).

I solved it though, guess just got stuck for a minute.
 
Usually people want the derivative of y wrt x. Plus, you gave an equation for y in terms of x and y.
 
Since y does not equal f(x), y can not be expressed in terms of x. We would have to express x in terms of y.

Of course, it would be much better if the equation was x=y^x, then it can be expressed as y=x^(1/x).
 

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